What is a Gage R&R ?

Gage R&R focuses on two key aspects of measurement:

Repeatability: Repeatability is the variation between successive measurements of the same part or trait by the same person using the same gage. In other words, how much variation do we see in measurements taken by the same person, on the same part, using the same tool?

Reproducibility: Reproducibility is the difference in the average of the measurements made by different people using the same instrument when measuring the identical characteristics on the same part. In other words, how much variation do we see in measurements taken by different people on the same part using the same tool?

(Ted Hessing, R.P. (2024) Gage repeatability and reproducibility (R&R))

What type of Gage R&R did we do ?

In our analysis we conducted a Crossed Gage R&R

  • Crossed-gage R&R is when each operator measures each part, and it must have a balanced design with random factors. It is used for non-destructive testing.

  • In this Gage R&R study, the measurement system analysis was conducted using Vernier calipers, to evaluate repeatability and reproducibility among operators and equipment.

Part Description

Product description

  • Single thread screw
  • 5 mm diameter
  • 50 mm length
  • Carbon steel (SAE 1022)
  • Application; structural timber products
  • Manufactured to EN14592:2008
  • Measurement Device : Vernier Caliper
  • Temperature: 21°C

Flow Chart

Flow chart Showing the Gage R&R process

Code

# Data for 10 parts, 3 operators and 2 measurements per part

Operator<- factor(rep(1:3, each = 20))

Part<- factor(rep(rep(1:10, each = 2), 3))

# When inputting #operator data: part 1, part 1, part 2, part 2 etc.

Diameter<-c(49.28, 49.27, 49.36, 49.42, 49.26, 49.25, 46.35, 46.36, 49.26, 49.24,#op1

            45.82, 45.81, 49.25, 49.27, 47.45, 47.45, 49.38, 49.37, 49.40, 49.44,

            ##########################

            49.28, 49.35, 49.40, 49.37, 49.26, 49.26, 46.39, 46.41, 49.27, 49.29,#op2

            45.75, 45.86, 49.30, 49.35, 47.47, 47.58, 49.39, 49.42, 49.44, 49.43,

            ##########################

            49.28, 49.27, 49.37, 49.39, 49.34, 49.34, 46.44, 46.42, 49.32, 49.31,#op3

            45.83, 45.91, 49.34, 49.28, 47.51, 47.54, 49.47, 49.36, 49.50, 49.45)

Data<-data.frame(Part,Operator,Diameter); print(Data)



#Load package

library("SixSigma")



#Perform gage R & R

ss.rr(var = Diameter, part = Part, appr = Operator, data = Data,

              

               main = "Six Sigma Gage R&R Study",

               sub = "",

               alphaLim = 0.05,

               errorTerm = "interaction",

               digits = 4,

               method = "crossed",

               print_plot = TRUE,

               signifstars = TRUE)

Results

##    Part Operator Diameter
## 1     1        1    49.28
## 2     1        1    49.27
## 3     2        1    49.36
## 4     2        1    49.42
## 5     3        1    49.26
## 6     3        1    49.25
## 7     4        1    46.35
## 8     4        1    46.36
## 9     5        1    49.26
## 10    5        1    49.24
## 11    6        1    45.82
## 12    6        1    45.81
## 13    7        1    49.25
## 14    7        1    49.27
## 15    8        1    47.45
## 16    8        1    47.45
## 17    9        1    49.38
## 18    9        1    49.37
## 19   10        1    49.40
## 20   10        1    49.44
## 21    1        2    49.28
## 22    1        2    49.35
## 23    2        2    49.40
## 24    2        2    49.37
## 25    3        2    49.26
## 26    3        2    49.26
## 27    4        2    46.39
## 28    4        2    46.41
## 29    5        2    49.27
## 30    5        2    49.29
## 31    6        2    45.75
## 32    6        2    45.86
## 33    7        2    49.30
## 34    7        2    49.35
## 35    8        2    47.47
## 36    8        2    47.58
## 37    9        2    49.39
## 38    9        2    49.42
## 39   10        2    49.44
## 40   10        2    49.43
## 41    1        3    49.28
## 42    1        3    49.27
## 43    2        3    49.37
## 44    2        3    49.39
## 45    3        3    49.34
## 46    3        3    49.34
## 47    4        3    46.44
## 48    4        3    46.42
## 49    5        3    49.32
## 50    5        3    49.31
## 51    6        3    45.83
## 52    6        3    45.91
## 53    7        3    49.34
## 54    7        3    49.28
## 55    8        3    47.51
## 56    8        3    47.54
## 57    9        3    49.47
## 58    9        3    49.36
## 59   10        3    49.50
## 60   10        3    49.45
## Complete model (with interaction):
## 
##               Df Sum Sq Mean Sq  F value   Pr(>F)    
## Part           9 105.15  11.683 11076.17  < 2e-16 ***
## Operator       2   0.02   0.012    11.51 0.000603 ***
## Part:Operator 18   0.02   0.001     0.94 0.543185    
## Repeatability 30   0.03   0.001                      
## Total         59 105.23                              
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## alpha for removing interaction: 0.05 
## 
## 
## Reduced model (without interaction):
## 
##               Df Sum Sq Mean Sq  F value   Pr(>F)    
## Part           9 105.15  11.683 10654.15  < 2e-16 ***
## Operator       2   0.02   0.012    11.07 0.000111 ***
## Repeatability 48   0.05   0.001                      
## Total         59 105.23                              
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Gage R&R
## 
##                        VarComp %Contrib
## Total Gage R&R    0.0016487674     0.08
##   Repeatability   0.0010965972     0.06
##   Reproducibility 0.0005521701     0.03
##     Operator      0.0005521701     0.03
## Part-To-Part      1.9470351659    99.92
## Total Variation   1.9486839333   100.00
## 
##                       StdDev  StudyVar %StudyVar
## Total Gage R&R    0.04060502 0.2436301      2.91
##   Repeatability   0.03311491 0.1986895      2.37
##   Reproducibility 0.02349830 0.1409898      1.68
##     Operator      0.02349830 0.1409898      1.68
## Part-To-Part      1.39536202 8.3721721     99.96
## Total Variation   1.39595270 8.3757162    100.00
## 
## Number of Distinct Categories = 48

Interpration of Results

Gage R&R Acceptance Criteria

(Interpreting Minitab’s Gage R&R chart - business performance improvement (BPI) (2023)

  • Total Gage R&R % Contribution: 0.08% (Below 1% → Acceptable)

  • Total Gage R&R % Study Variation: 2.91% (Below 10% → Acceptable)

  • Number of Distinct Categories (NDC): 48 (Greater than 10 → Acceptable)

  • Part-to-Part Variation: 99.92% (Ideal—most variation comes from actual part differences).

  • Operator Effect: Statistically significant (p = 0.000111) but practically small (0.03% contribution).

Variation of Components

  • Part2Part Variation at 100% is high showing that the measurement system recognize the variation of each part across 3 different operators.

  • Gage R&R , Repeatability and Reproduciblty are all low indicating minimal error amongst the operator and the equipment

Comparison of Operator Results

  • The results there was very little variation in result from each operator showing that the measurement technique can be seen as consistent and accurate supporting the result we have obtained from the Gage R&R study.

CAP Study

library(qcc) 

# Data from the provided 10 results
cap_data <- c(49.28, 49.36, 49.26, 46.35, 49.26, 
              45.82, 49.25, 47.45, 49.38, 49.40)

# Create a qcc object for process capability analysis
xbar <- qcc(cap_data, type = "xbar.one", title = "Capability Study")
# Process Capability Analysis
process.capability(xbar, spec.limits = c(47, 51), target = 50)

Results 

## 
## Process Capability Analysis
## 
## Call:
## process.capability(object = xbar, spec.limits = c(47, 51), target = 50)
## 
## Number of obs = 10           Target = 50
##        Center = 48.48           LSL = 47
##        StdDev = 1.637           USL = 51
## 
## Capability indices:
## 
##        Value     2.5%   97.5%
## Cp    0.4072  0.22306  0.5920
## Cp_l  0.3015  0.09243  0.5107
## Cp_u  0.5129  0.24907  0.7767
## Cp_k  0.3015  0.05237  0.5507
## Cpm   0.2985  0.14501  0.4532
## 
## Exp<LSL 18%   Obs<LSL 20%
## Exp>USL 6.2%  Obs>USL 0%

Process Capability Summary

  • Low Capability (Cp = 0.4072) → High variation, not within specs.
  • Off-Target (Cp_k = 0.3015) → Mean not centered at 50.
  • High Defects: 20% below LSL, 6.2% expected above USL.

Reflection

(Drroopesh and Drroopesh (2025) Reflection)

Declaration

DECLARATION: I declare that:

  • This work is entirely my own, and no part of it has been copied from any other person’s words or ideas, except as specifically acknowledged through the use of inverted commas and in-text references;

  • No part of this assignment has been written for me by any other person except where such collaboration has been authorised by the lecturer concerned;

  • I understand that I am bound by DkIT Academic Integrity Policy. I understand that I may be penalised if I have violated the policy in any way;

  • I have not used generative artificial intelligence (AI) (e.g. ChatGPT) unless it has been permitted by the lecturer(s) concerned;

  • This assignment has not been submitted for any other module at DkIT or any other institution, unless authorised by the relevant Lecturer(s);

  • I have read and abided by all of the requirements set down for this assignment.

TYPED SIGNATURE:Niall O’Callaghan

DATE: 03/03/25

Refernces